The metric dimension has been introduced independently by Harary, Melter and Slater in 1975 to identify vertices of a graph G using its distances to a subset of vertices of G. A resolving set X of a graph G is a subset of vertices such that, for every pair (u,v) of vertices of G, there is a vertex x in X such that the distance between x and u and the distance between x and v are distinct. The metric dimension of the graph is the minimum size of a resolving set. Computing the metric dimension of a graph is NP-hard even on split graphs and interval graphs. Bonnet and Purohit proved that the metric dimension problem is W[1]-hard parameterized by treewidth. Li and Pilipczuk strenghtened this result by showing that it is NP-hard for graphs of treewidth. In this article, we prove that that metric dimension is FPT parameterized by treewidth in chordal graphs.

翻译：度量维数由Harary、Melter和Slater于1975年独立提出，通过图G中顶点到其子集的距离来识别顶点。图G的解析集X是一个顶点子集，满足：对G中任意一对顶点(u,v)，存在X中的顶点x，使得x到u的距离与x到v的距离不同。图的度量维数是最小解析集的大小。即使在分裂图和区间图上，计算图的度量维数也是NP难的。Bonnet与Purohit证明度量维数问题在树宽参数化下是W[1]-难的。Li与Pilipczuk通过证明该问题对于具有树宽的图是NP难的，强化了这一结论。本文证明弦图中度量维数问题在树宽参数化下属于固定参数易解类（FPT）。